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Maximal injective subalgebras in factors associated with free groups. (English) Zbl 0545.46041
In this paper concrete examples of maximal injective von Neumann algebras in type \(II_ 1\) factors are constructed. In particular if the free group of n generators \({\mathbb{F}}_ n\) acts freely on some nonatomic probability space \((X,\mu)\) by measure preserving automorphisms and M denotes the associated group measure algebra and \(R_ u\) denotes the injective subalgebra of M corresponding to the action of the generator \(u\in {\mathbb{F}}_ n\) on \((X,\mu)\), then \(R_ u\) is a maximal injective von Neumann subalgebra of M. The subalgebra \(R_ u\) can be any injective type \(II_ 1\) von Neumann algebra provided with a suitable action of \({\mathbb{F}}_ n\) on \((X,\mu)\).
Two old problems of R. V. Kadison on the embedding of the hyperfinite factor \(R_ u\) posed in the Baton Rouge Conference are also solved.
Reviewer: V.I.Ovchinnikov

MSC:
46L10 General theory of von Neumann algebras
46L35 Classifications of \(C^*\)-algebras
22D15 Group algebras of locally compact groups
46L55 Noncommutative dynamical systems
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